The mode of a continuous random variable having density f  is the value of x for which f(x...

The mode of a continuous random variable having density f  is the value of x for which f(x) attains its maximum. Compute the mode of X in cases (a), (b), and (c) of Theoretical Exercise 1.

Exercise 1

The median of a continuous random variable having distribution function F is that value m such that F(m)= . That is, a random variable is just as likely to be larger than its median as it is to be smaller. Find the median of X if X is

(a) uniformly distributed over (a, b);

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(b) normal with parameters μ,σ2;

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(c) exponential with rate λ.